241lecture 10 - Lecture notes 10 PDF

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Fall 2020

21-241, Section 5

Page 1 of 3

21-241: Matrices and Linear Transformations, Lecture 5 Lecture #10 Outline Read: Section 3.B of LADR (Linear Algebra Done Right) Null Spaces and Ranges Definition: Let T ∈ L(V,W ). Then null(T ) = {v ∈ V | T v = 0}. null(T ) is also called the kernel of T.

Theorem 3.14: The kernel of T ∈ L(V,W ) is a subspace of V .

Definition: Let T : V → W . We say that T is injective or one-to-one if T u = T v implies u = v .

Theorem 3.16: Let T ∈ L(V,W ). Then T is injective if and only if null(T ) = {0}. Proof:

Definition: Let T : V → W . The range of T is the set range(T ) = {T v | v ∈ V } ⊂ W . We say that T is surjective or onto W if range(T ) = W .

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Fall 2020

21-241, Section 5

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Theorem 3.19: Let T ∈ L(V,W ). Then range(T ) is a subspace of W .

Theorem 3.22 (Fundamental Theorem of Linear Maps): Suppose V is a finite-dimensional vector space and T ∈ L(V,W ). Then rangeT is finite-dimensional and dim V = dim nullT + dim rangeT . Proof:

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Fall 2020

21-241, Section 5

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Corollary 3.23: Suppose V and W are finite-dimensional vector spaces such that dim V > dim W . Then no linear map from V to W is injective.

Corollary 3.24: Suppose V and W are finite-dimensional vector spaces such that dimV < dim W . Then no linear map from V to W is surjective.     x1  x1 − x3 + 2x4    x     2 Example: Let T : R4 → R3 be defined by T    = 2x1 + x2 − x4 . Determine if T is injective and/or  x3  x2 + 3x3 − x4 x4 surjective.

Corollary 3.26: A homogeneous system of linear equations with more variables than equations has nonzero solutions.

Corollary 3.29: An inhomogeneous system of linear equations with more equations than variables has no solution for some choice of the constant terms. Exercises 1. LADR Section 3.B # 1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 14, 17, 18, 19, 22, 23, 26

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